CHAPTER 2: INTRODUCTION TO VARIABLES AND PROPERTIES OF ALGEBRA

Transcription

1 CHAPTER 2: INTRODUCTION TO VARIABLES AND PROPERTIES OF ALGEBRA Chapter Objectives By the end of this chapter, students should be able to: Interpret different meanings of variables Evaluate algebraic expressions Writing algebraic expressions Identify properties of algebra Commutative Associative Identity Inverse Distributive Contents CHAPTER 2: INTRODUCTION TO VARIABLES AND PROPERTIES OF ALGEBRA SECTION 2.1 INTRODUCTION TO VARIABLES A. WHAT IS A VARIABLE? B. MEANING OF A VARIABLE IN MATHEMATICS C. VARIABLE EXPRESSIONS D. WRITING ALGEBRAIC EXPRESSIONS E. EVALUATE ALGEBRAIC EXPRESSIONS F. LIKE TERMS AND COMBINE LIKE TERMS EXERCISES SECTION 2.2 PROPERTIES OF ALGEBRA EXERCISES CHAPTER REVIEW

2 SECTION 2.1 INTRODUCTION TO VARIABLES A. WHAT IS A VARIABLE? When someone is having trouble with algebra, they may say, I don t speak math! While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a like a language that you must learn in order to be successful. In order to understand math, you must practice the language. Action words, like run, jump, or drive, are called verbs. In mathematics, operations are like verbs because they involve doing something. Some operations are familiar, such as addition, multiplication, subtraction, or division. Operations can also be much more complex like a raising to an exponent or taking a square root. A variable is one of the most important concepts of mathematics, without variables we would not get far in this study. A variable is a symbol, usually an English letter, written to replace an unknown or changing quantity. Definitions A variable, usually represented by a letter or symbol, can be defined as: A quantity that may change within the context of a mathematical problem. A placeholder for a specific value. An algebraic expression is a mathematical statement that can contain numbers, variables, and operations (addition, subtraction, multiplication, division, etc ). MEDIA LESSON Introduction to variables and variable expressions (Duration 7:54) View the video lesson, take notes and complete the problems below. Definition: A is a that represents an. a) How many hours will you study tomorrow? b) How much will you pay to have your car repaired? A consists of,, and like,, and. Often and are used. The difference is a contain a variable while the. 90

3 Mathematical Expressions Equations Writing variable expressions a) It costs $9 per adult and $5 per child to go to the movies. Variable expression for the cost of a group go to the movies: b) It costs $30 per day to rent a car plus $0.10 per mile. Variable expression for the total rental cost: Evaluating variable expressions c) Evaluate 5x + 7 when x = 6 d) Evaluate 4m 3n when m = 9 and n = 7 e) Evaluate p 2 3q + 7 when p = 11, q = 8 f) Evaluate 36 d and f = 1 +7e 9f when d = 9, e = 3, MEDIA LESSON Why all the letters in algebra? (Duration 3:03) View the video lesson, take notes and complete the problems below. What question do students ask a lot when they start Algebra? What are other things besides letters that can be used as a variable? 91

4 B. MEANING OF A VARIABLE IN MATHEMATICS MEDIA LESSON Meaning of a variable (Duration 4:13) View the video lesson, take notes and complete the problems below. Definitions of a variable 1) Define variable as a changing value A variable is a letter that represents or that may within the context of a mathematical problem. Example: 2) Define variable as a placeholder A variable is a letter that represents or that will remain based on the confines of the mathematical problem. Example: Example: Determine if the description describes a changing value (CV) or a placeholder (P) then determine a possible variable. Changing Value (CV) or Scenario Variable Placeholder (P) The elevation of Mount Humphrey s The water level of a pool as it is being filled The number of calories consumed throughout the day The monthly car payment with a fixed interest loan The amount of gas consumed by your car as you drive The cost of a new textbook for a specific class from the bookstore at the beginning of the semester YOU TRY Scenario The number of cars in the parking lot at this moment The number of cars pass by an intersection throughout the day The altitude of an airplane during a trip The temperature in the Dead Valley on at midnight on January 1 st in 1905 The balance of your checking account today Changing Value (CV) or Placeholder (P) Variable 92

5 MEDIA LESSON Why aren't we using the multiplication sign? (Duration 3:08) View the video lesson, take notes and complete the problems below. Why the multiplication sign x is not being used much when we get to algebra? MEDIA LESSON Why is 'x' the symbol for an unknown? (Duration 3:57) View the video lesson, take notes and complete the problems below. Why is it that x is the unknown? C. VARIABLE EXPRESSIONS In mathematics, especially in algebra, we look for patterns in the numbers that we see. Using mathematical verbs and variables studied in previous lessons, expressions can be written to describe a pattern. Definition An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations. An algebraic expression consists of coefficients, variables, and terms. Given an algebraic expression, a coefficient is the number in front of the variable. variable is a letter representing any number. term is a product of a coefficient and variable(s). constant: a number with no variable attached. A term whose value never changes. For example, t, 2x, 3st, 7x 2, 5ab 3 c are all examples of terms because each is a product of a coefficient and variable(s). 5xy 2 term 2x term 3 constant term MEDIA LESSON Algebraic expression vocabulary (Duration 5:52) View the video lesson, take notes and complete the problems below. Terms:. Constant Term:. Example 1: Consider the algebraic expression 4x 5 + 3x 4 22x 2 x + 17 a) List the terms: b) Identify the constant term: Factors:. 93

6 Coefficient:. Example 2: Complete the table below. List the Factors Identify the Coefficient 4m x 1 2 bh 2r 5 Example 3: Consider the algebraic expression 5y 4 8y 3 + y 2 y 7. 4 a) How many terms are there? b) Identify the constant term. c) What is the coefficient of the first term? d) What is the coefficient of the second term? e) What is the coefficient of the third term? f) List the factors of the fourth term._ YOU TRY Consider the algebraic expression 2m 3 + m 2 2m 8. a) How many terms are there? b) Identify the constant term. c) What is the coefficient of the first term? d) What is the coefficient of the second term? e) List the factors of the third term. D. WRITING ALGEBRAIC EXPRESSIONS MEDIA LESSON Write algebraic expressions (Duration 6:18) View the video lesson, take notes and complete the problems below. Example 1: Juan is 6 inches taller than Niko. Let N represent Niko s height in inches. Write an algebraic expression to represent Juan s height. Niko s height: Juan s height: Example 2: Juan is 6 inches taller than Niko. Let J represent Juan s height in inches. Write an algebraic expression to represent Niko s height. Niko s height: Juan s height: 94

7 Example 3: Suppose sales tax in your town is currently 9.8%. Write an algebraic expression representing the sales tax for an item that costs D dollars. Cost: Sales tax: Example 4: You started this year with $362 saved and you continue to save an additional $30 per month. Write an algebraic expression to represent the total amount saved after m months. Number of months: Total amount saved: Example 5: Movie tickets cost $8 for adults and $5.50 for children. Write an algebraic expression to represent the total cost for A adults and C children to go to a movie. Number of adults: Number of children: Total Cost: YOU TRY Complete the following problems. Show all steps as in the media examples. a) There are about 80 calories in one chocolate chip cookie. If we let n be the number of chocolate chip cookies eaten, write an algebraic expression for the number of calories consumed. Number of cookies: Number of calories consumed: b) Brendan recently hired a contractor to do some necessary repair work. The contractor gave a quote of $450 for materials and supplies plus $38 an hour for labor. Write an algebraic expression to represent the total cost for the repairs if the contractor works for h hours. Number of hours worked: Total cost: c) A concession stand charges $3.50 for a slice of pizza and $1.50 for a soda. Write an algebraic expression to represent the total cost for P slices of pizza and S sodas. Number of slices of pizza: Number of sodas: Total cost: 95

8 MEDIA LESSON The story of x (Duration 7:09) View the video lesson, take notes and complete the problems below. Example 1: Tell the story of x in each of the following expressions. a) x 5 b) 5 x c) 2x d) x 2 Example 2: Tell the story of x in each of the following expressions. a) 2x + 4 b) 2(x + 4) c) 5(x 3) 2 2 Example 3: Write an algebraic expression that summarizes the stories below. a) Step 1: Add 3 to x Step 2: Divide by 2 b) Step 1: Divide x by 2 Step 2: Add 3 Example 4: Write an algebraic expression that summarizes the stories below. Step 1: Subtract x from 7 Step 2: Raise to the third power Step 3: Multiply by 3 Step 4: Add 1 96

9 Below is a table of common English words converted into a mathematical expression. You can use this table to assist in translating expressions. Operation Words Example Translation Added to 4 added to n n + 4 More than 2 more than y y + 2 Addition The sum of The sum of r and s r + s Increased by m increased by 6 m + 6 The total of The total of 8 and x 8 + x Plus c plus 2 c + 2 Minus x minus 1 x 1 Less than 5 less than y y 5 Subtraction Multiplication Less 4 less r 4 r Subtracted from 3 subtracted from t t 3 Decreased by m decreased by 10 m 10 The difference between The difference between x and y x y Times 12 times x 12 x Of One-third of v 1 3 v The product of The product of n and k nk or n k Multiplied by y multiplied by 3 3y Division Division Power Equals Twice Twice d 2d or 2 d Divided by n divided by 4 The quotient of The ratio of Per The quotient of t and x The ratio of x to p 2 per b The square of The square of y y 2 The cube of The cube of k k 3 Is Are Equal Gives Is equal to Is equivalent to Yields Results in was n 4 t x x p 2 b 97

10 YOU TRY Complete the following problems. a) Tell the story of x in the expression x 3 5. b) Write an algebraic expression that summarizes the story below: Step 1: Multiply x by 2 Step 2: Add 5 Step 3: Raise to the second power. MEDIA LESSON The beauty of algebra (Duration 10:06) View the video lesson, take notes and complete the problems below. What algebraic expression did Sal start with when he discussed why the abstraction of mathematics is so fundamental? E. EVALUATE ALGEBRAIC EXPRESSIONS MEDIA LESSON Evaluate algebraic expressions (Duration 9:33) View the video lesson, take notes and complete the problems below To evaluate a algebraic or variable expression: PEMDAS P: E: MD: AS: 98

11 Example 1: Find the value of each expression when w = 2. Simplify your answers. a) w 6 b) 6 w c) 5w 3 d) w 3 e) 3w 2 f) (3w) 2 g) 4 5w h) 5w 4 i) 3 w Example 2: Evaluate ab + c given a = 5, b = 7, and c = 3. Example 3: Evaluate a 2 b 2 given a = 5 and b = 3. Example 4: A local window washing company charges $11.92 for each window plus a reservation fee of $7. a) Write an algebraic expression to represent the total cost from the window washing company for washing w windows. b) Use this expression to determine the total cost for washing 17 windows. YOU TRY Given a = 5, b = 1, c = 2, evaluate the expressions below. Show all steps. a) Evaluate b 2 4ac. b) Evaluate 2a 5b + 7c. 99

12 F. LIKE TERMS AND COMBINE LIKE TERMS Definition Two terms are like terms if the base variable(s) and exponent on each variable are identical. For example, 3x 2 y and 7x 2 y are like terms because they both contain the same base variables, x and y, and the exponents on x (the x is squared on both terms) and y are the same. Combining like terms: If two terms are like terms, we add (or subtract) the coefficients, then keep the variables (and exponents on the corresponding variable) the same. MEDIA LESSON Like terms and combine like terms (Duration 6:30) View the video lesson, take notes and complete the problems below. Example 1: Identify the like terms in each of the following expressions. 3a 6a + 10a a 5x 10y + 6z 3x 7n + 3n 2 2n 3 + 8n 2 + n n 3 Example 2: Combine the like terms. 3a 6a + 10a a 5x 10y + 6z 3x 7n + 3n 2 2n 3 + 8n 2 + n n 3 YOU TRY Combine the like terms. a) 3x 4x + x 8x b) 5 + 2a 2 4a + a

13 EXERCISES Tell the story of x in each of the following expressions. 1) x 11 2) x + 5 3) 5x 4) x 5 5) x 3 6) 2 x 7) 2x 3 8) 8x 2 9) (2x) 2 10) 7 2x 11) 5(7 x) 3 12) ( 3x 3 5 )3 Write an algebraic expression that summarizes the stories below. 13) Step 1: Add 8 to x 14) Step 1: Divide x by 8 Step 2: Raise to the third power Step 2: Subtract 5 15) Step 1: Subtract 3 from x Step 2: Multiply by 7 16) Step 1: Multiply x by 10 Step 2: Raise to the 3rd power Step 3: Multiply by 2 17) Step 1: Add 5 to x Step 2: Divide by 2 Step 3: Raise to the second power Step 4: Add 8 19) Step 1: Subtract x from 2 Step 2: Multiply by 8 Step 3: Raise to the third power Step 4: Add 1 Step 5: Divide by 3 18) Step 1: Raise x to the second power Step 2: Multiply by 5 Step 3: Subtract from 9 20) Step 1: Multiply x by 4 Step 2: Add 9 Step 3: Divide by 2 Step 4: Raise to the fifth power Find the value of each expression when b = 8. Simplify your answers. 21) b 11 22) b ) 5b 24) b 2 25) b 3 26) 2 b Evaluate each of the following given q = ) 2q 3 28) 8q 2 29) (2q) 2 30) 4 7q 31) 7 2q 32) 2 q 101

14 Evaluate the following expressions for the given values. Simplify your answers. 33) b for a = 6, b = 4 2a 35) 3 5 ab for a = 8, b = x 8 34) for x = 3 5+x 36) 3x 2 + 2x 1 for x = 1 37) x 2 y 2 for x = 3, y = 2 38) 2x 7y for x = 5, y = 3 39) Shea bought c candy bars for $1.50 each. Write an algebraic expression for the total amount Shea spent. 40) Suppose sales tax in your town is currently 9%. Write an algebraic expression representing the sales tax for an item that costs d dollars. 41) Ben bought m movie tickets for $8.50 each and p bags of popcorn for $3.50 each. a) Write an algebraic expression for the total amount Ben spent. b) Use this expression to determine the amount Ben will spend if he buys 6 movie tickets and 4 bags of popcorn. 42) Noelle is 5 inches shorter than Amy. Amy is A inches tall. Write an algebraic expression for Noelle's height. 43) Jamaal studied h hours for a big test. Karla studied one fourth as long. Write an algebraic expression for the length of time that Karla studied. 44) A caterer charges a delivery fee of $45 plus $6.50 per guest. a) Write an algebraic expression to represent the total catering cost if g guests attend the reception. b) Use the expression to determine the amount of a company luncheon of 50 guests. 45) Tickets to the museum cost $18 for adults and $12.50 for children. a) Write an algebraic expression to represent the cost for a adults and c children to visit the museum. b) Use this expression to determine the cost for 4 adults and 6 children to attend the museum. 46) Consider the algebraic expression 5n 8 n 5 + n 2 + n 8 2 a) How many terms are there? b) Identify the constant term. c) What is the coefficient of the first term? d) What is the coefficient of the second term? e) What is the coefficient of the third term? f) What is the coefficient of the fourth term? Combine the like terms. 47) 3d 5d + d 7d 48) 3x 2 + 3x 3 9x 2 + x x 3 49) a 2b + 4a + b ( 2b) 50) 3x 7y + 9x 5y 7 Log on to Canvas to take the section quiz 102

15 SECTION 2.2 PROPERTIES OF ALGEBRA Properties of real numbers are the basic rules when we work with expressions in Algebra. Addition a + 0 = a Multiplication a 1 = a Identity Hint: the number stays true to its identity. Any number plus zero is the same number = 7 x + 0 = x + 0 = Any number multiplies by one is the same number. 7 1 = 7 x 1 = x 1 = Inverse Hint: Inverse reverse undo Additive Inverse a + ( a) = 0 Any number plus its opposite equals zero. 2 + ( 2) = 0 x + ( x) = 0 Multiplicative inverse a 1 a = 1 if a is not zero Any number multiplied by its reciprocal is one = 1 x 1 x = 1 Commutative Hint: Commute move switch places a + b = b + a You can add in any order = x + 5 = 5 + x + = + a b = b a You can multiply in any order. 2 3 = 3 2 x 2 = 2 x = Associative Hint: associate different groups parenthesis (a + b) + c = a + (b + c) When you add, you can group in any combination. (3 + 5) + 2 = 3 + (5 + 2) (x + y) + z = x + (y + z) (a b) c = a (b c) When you multiply, you can group in any combination. (6 2) 7 = 6 (2 7) (x y) z = x (y z) Distributive Hint: Distribute give out to Multiplying a number by a group of numbers added together or subtracted each other is the same as doing each multiplication separately. 5( ) = (x + y) = 2x + 2y 103

16 MEDIA LESSON Properties of Real Numbers (Duration 9:59) View the video lesson, take notes and complete the problems below. The commutative properties The associative properties Identity properties Inverse properties 104

17 Distributive property NOTE: Subtraction and division do not have the associative and communicative properties. 105

18 EXERCISES Identify the property that justifies each problem below. 1) w + 0 = w Name: 2) x + x = 0 3) 3(u + v) = 3u + 3v Name: Name: 4) 5w(6z) = (6z)5w Name: 5) (5) 1 5 x = 1x Name: 6) 2 (4a)b = (2 4)ab Name: 7) (5x + 5) + 4 = 5x + (5 + 4) Name: 8) 2x(3y + 7z) = 2x(7z + 3y) Name: Find the additive inverse and the multiplicative inverse for each problem below. Additive inverse Multiplicative inverse 9) 6 10) ) ) m (given m 0) Complete the following table by performing the indicated operations by computing the result in the parentheses first as the order of operations requires. Problem 1 Problem 2 Are the Results the Same? (5 + 7) (7 + 3) 13) Addition _ (10 5) 4 10 (5 4) 14) Subtraction _ (2 3) 4 2 (3 4) 15) Multiplication _ (600 30) (30 5) 16) Division _ 106

19 Use the commutative or associative properties to perform the operations in the order you find most simple. Show all your work. 17) ( ) ) 5 (6 8) 19) (15 + 4) ) Apply distributive property and combine like terms if possible. 21) 8n(5 m) 22) 7k( 6x + 6) 23) 6(1 + 6x) 24) 8x(5 + 10n) 25) ( 5 + 9a) 26) (n + 1)2 27) 4(x + 7) + 8(x + 4) 28) 8(n + 6) 8(n + 8) 29) 8x + 9( 9x + 9) 30) 4v 7(1 8v) 31) 10(x 2) 3 32) 7(7 + 3v) + 10(3 10v) 33) (x 2 8) (2x 2 7) 34) 9(b + 10) + 5b 35) 2n( 10n + 5) 7(6 10n 3m) Log on to Canvas to take the section quiz 107

20 CHAPTER REVIEW KEY TERMS AND CONCEPTS Look for the following terms and concepts as you work through the workbook. In the space below, explain the meaning of each of these concepts and terms in your own words. Provide examples that are not identical to those in the text or in the media lesson. Variable Algebraic expression Coefficient Term Constant Like terms Identity property of addition Identity property of multiplication Additive Inverse property Multiplicative Inverse property Associative property of addition Associative property of multiplication Commutative property of addition Commutative property of multiplication Distributive property 108

21 Tell the story of x in each of the following expressions. 1) x ) 4(x 2 + 3) 3) x+3 5 Write an algebraic expression that summarizes the stories below. 4) Step 1: Add 2 to x Step 2: Raise to the second power Step 3: Divide by 3 5) Step 1: Subtract 2 from x Step 2: Divide by 2 Step 3: Add 1 Step 4: Divide by 3 Evaluate the following expressions when a = 4, b = 2, and c = 1. Simplify your answers. 6) c 3 9) b 7) 6 b +c 2 c 3 10) a 2 + 2a ac 8) a b c 11) c b 12) Suppose sales tax in your city is 8.25%. Write an algebraic expression representing the sales tax for an item that cost x dollars. 13) Will bought c candy bars for $1.25 each. Write an algebraic expression for the total amount Will spent. 14) Consider the algebraic expression 2x + 3x 2 x 2 x13 a) How many terms are there? b) Identify the constant terms. c) What is the coefficient of the first term? d) What is the coefficient of the second term? e) What is the coefficient of the third term? Combine the like terms of the following expressions. 15) 2x 3x + 4y 16) 2x + y 1 y 5x 17) 1x + 2x x 2 + 3x 2x 2 18) 4x 2y + 3x + 4y + 2xy Simplify. Combine like terms when possible. 19) ( ) ) 4 (12 2) 21) n(5 2) 22) (n + 1)2 + n ) 9n (n + 1) + 7n 24) 2y + y(y + y) 25) z(z + 1) z ) 1 (2x 4) ) 2x+4 +10(2x ) 109

22 Find the additive and multiplicative inverse of the following problems. 28) 10 29) ) x 1 110